Shame about council tax. Do you have any other letters from the government dated in that month? (eg DVLA, HMRC, DWP, electoral register)
In case you don't, it's useful to read the following:
From
https://assets.publishing.service.gov.u ... retion.pdf page 15:
You must check the available evidence to see whether an applicant meets the residence requirements.
The following can be used as evidence of residence:
• passports or travel documents which have been stamped to show arrival in the UK and entry and departure from other countries: these should be checked against the list of absences that applicants are asked to provide on the application form
• Home Office records
• if the applicant does not have passports to cover the qualifying period, other
evidence such as employers’ letters or tax and National Insurance letters:
o in such cases you should assess whether there is sufficient evidence to
show that that applicant has been resident in the UK during the qualifying period, giving them the benefit of any doubt where claimed absences are within the limits we would normally allow and there are no grounds to doubt the accuracy of the claim
You must not normally accept doctors' letters on their own as proof of residence. However, if nothing else is available and the doctors can confirm that they have seen the applicant on a regular basis during the period concerned these may be accepted.
If there are gaps in a person’s evidence of residence and it is clear from the information available that they could not have travelled, you must accept this. Examples of this might include a refugee who has no means of travel or where immigration records confirm continuous residence.
Therefore UKVI records should be fine as they fall under Home Office. Should this period of unemployment have happened during lockdown, the above would also be enough to justify you were in the country as you couldn't have travelled.
All advice comes from personal research and experience and should not be regarded as professional opinion.
